arXiv research

The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.

The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.

We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.

We present a framework to derive risk bounds for vector-valued learning with a broad class of feature maps and loss functions. Multi-task learning and one-vs-all multi-category learning are treated as examples. We discuss in detail vector-valued functions with one hidden layer, and demonstrate that the conditions under…

Optimal rates for vector-valued regression on various norms.

Study confirms learning rates for vector-valued spectral algorithms, proving consistency.

The paper shows vector-valued risk measures ignore dependence structures.

Develops vector-valued RKBS for neural networks and operators.

Vector-valued learning, where the output space admits a vector-valued structure, is an important problem that covers a broad family of important domains, e.g. multi-task learning and transfer learning. Using local Rademacher complexity and unlabeled data, we derive novel semi-supervised excess risk bounds for general v…

Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

Paper analyzes error bounds for learning with vector-valued RF, improving existing analyses.

Randomized algorithm solves vector-valued regression problems with low-rank operators.

Paper introduces vector-valued variation spaces for multi-output neural networks.

Study optimizes online learning for vector-valued data regression.

Multi-output Gaussian processes (MOGP) are probability distributions over vector-valued functions, and have been previously used for multi-output regression and for multi-class classification. A less explored facet of the multi-output Gaussian process is that it can be used as a generative model for vector-valued rando…

We discuss sharp Sobolev inequalities for vector valued maps.

Paper proposes a new method to evaluate joint risk under uncertainty.

A market model with $d$ assets in discrete time is considered where trades are subject to proportional transaction costs given via bid-ask spreads, while the existence of a numèraire is not assumed. It is shown that robust no arbitrage holds if, and only if, there exists a Pareto solution for some vector-valued utility…

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RK…

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

Boosting framework for vector-valued prediction with geometric stability.

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different char…

This paper presents a general vector-valued reproducing kernel Hilbert spaces (RKHS) framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space. Our formulation encompasses both Vector-valued Manifold Regularization and Co-regularized Multi-view…

Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.

Whitney type examples of maps $f\in C^k(\real^m,\real^n)$ for a maximal possible real $k$, and multidimensional space-filling curves with special properties are constructed.

In computer vision, image datasets used for classification are naturally associated with multiple labels and comprised of multiple views, because each image may contain several objects (e.g. pedestrian, bicycle and tree) and is properly characterized by multiple visual features (e.g. color, texture and shape). Currentl…

Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.

We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justificatio…

New kernels capture both local and non-local interactions efficiently.

New algorithms optimize multiple tasks with shared similarities, reducing regret.

Study vector-valued robust control under uncertainty.

Gaussian processes adapted for Riemannian manifolds using gauge-independent kernels.

In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

New method transfers emotions in facial images.

In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for discrete probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions given for continuous random variables when adapt…

In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics…

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical results for the minimal surface equation can be generalized. We also conjecture…

Nyström approximation for scalable operator learning

Study on identifying most preferred policy in bandits with vector-valued rewards.

This work studies the denoising of piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness over a graph, where the value at each node can be vector-valued. We extend the graph trend filtering framework to denoising vector-valued graph signals with a family of non-convex regularizers, which exhibi…

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…

Framework for transferring discount curve estimates across fixed-income product classes.

The contraction inequality for Rademacher averages is extended to Lipschitz functions with vector-valued domains, and it is also shown that in the bounding expression the Rademacher variables can be replaced by arbitrary iid symmetric and sub-gaussian variables. Example applications are given for multi-category learnin…

An $n$ dimensional minimal submanifold $Σ$ of $\R^{n+m}$ is called non-parametric if $Σ$ can be represented as the graph of a vector-valued function $f:D\subset \R^n \mapsto \R^m$. This note provides a sufficient condition for the stability of such $Σ$ in terms of the norm of the differential $df$.

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. T…

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…